Integrand size = 10, antiderivative size = 72 \[ \int \left (a \cosh ^2(x)\right )^{7/2} \, dx=\frac {16}{35} a^3 \sqrt {a \cosh ^2(x)} \tanh (x)+\frac {8}{35} a^2 \left (a \cosh ^2(x)\right )^{3/2} \tanh (x)+\frac {6}{35} a \left (a \cosh ^2(x)\right )^{5/2} \tanh (x)+\frac {1}{7} \left (a \cosh ^2(x)\right )^{7/2} \tanh (x) \]
8/35*a^2*(a*cosh(x)^2)^(3/2)*tanh(x)+6/35*a*(a*cosh(x)^2)^(5/2)*tanh(x)+1/ 7*(a*cosh(x)^2)^(7/2)*tanh(x)+16/35*a^3*(a*cosh(x)^2)^(1/2)*tanh(x)
Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.54 \[ \int \left (a \cosh ^2(x)\right )^{7/2} \, dx=\frac {1}{35} a^3 \sqrt {a \cosh ^2(x)} \left (35+35 \sinh ^2(x)+21 \sinh ^4(x)+5 \sinh ^6(x)\right ) \tanh (x) \]
Time = 0.46 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 3682, 3042, 3682, 3042, 3682, 3042, 3686, 3042, 3117}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \left (a \cosh ^2(x)\right )^{7/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a \sin \left (\frac {\pi }{2}+i x\right )^2\right )^{7/2}dx\) |
\(\Big \downarrow \) 3682 |
\(\displaystyle \frac {6}{7} a \int \left (a \cosh ^2(x)\right )^{5/2}dx+\frac {1}{7} \tanh (x) \left (a \cosh ^2(x)\right )^{7/2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \tanh (x) \left (a \cosh ^2(x)\right )^{7/2}+\frac {6}{7} a \int \left (a \sin \left (i x+\frac {\pi }{2}\right )^2\right )^{5/2}dx\) |
\(\Big \downarrow \) 3682 |
\(\displaystyle \frac {6}{7} a \left (\frac {4}{5} a \int \left (a \cosh ^2(x)\right )^{3/2}dx+\frac {1}{5} \tanh (x) \left (a \cosh ^2(x)\right )^{5/2}\right )+\frac {1}{7} \tanh (x) \left (a \cosh ^2(x)\right )^{7/2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \tanh (x) \left (a \cosh ^2(x)\right )^{7/2}+\frac {6}{7} a \left (\frac {1}{5} \tanh (x) \left (a \cosh ^2(x)\right )^{5/2}+\frac {4}{5} a \int \left (a \sin \left (i x+\frac {\pi }{2}\right )^2\right )^{3/2}dx\right )\) |
\(\Big \downarrow \) 3682 |
\(\displaystyle \frac {6}{7} a \left (\frac {4}{5} a \left (\frac {2}{3} a \int \sqrt {a \cosh ^2(x)}dx+\frac {1}{3} \tanh (x) \left (a \cosh ^2(x)\right )^{3/2}\right )+\frac {1}{5} \tanh (x) \left (a \cosh ^2(x)\right )^{5/2}\right )+\frac {1}{7} \tanh (x) \left (a \cosh ^2(x)\right )^{7/2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \tanh (x) \left (a \cosh ^2(x)\right )^{7/2}+\frac {6}{7} a \left (\frac {1}{5} \tanh (x) \left (a \cosh ^2(x)\right )^{5/2}+\frac {4}{5} a \left (\frac {1}{3} \tanh (x) \left (a \cosh ^2(x)\right )^{3/2}+\frac {2}{3} a \int \sqrt {a \sin \left (i x+\frac {\pi }{2}\right )^2}dx\right )\right )\) |
\(\Big \downarrow \) 3686 |
\(\displaystyle \frac {6}{7} a \left (\frac {4}{5} a \left (\frac {2}{3} a \text {sech}(x) \sqrt {a \cosh ^2(x)} \int \cosh (x)dx+\frac {1}{3} \tanh (x) \left (a \cosh ^2(x)\right )^{3/2}\right )+\frac {1}{5} \tanh (x) \left (a \cosh ^2(x)\right )^{5/2}\right )+\frac {1}{7} \tanh (x) \left (a \cosh ^2(x)\right )^{7/2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \tanh (x) \left (a \cosh ^2(x)\right )^{7/2}+\frac {6}{7} a \left (\frac {1}{5} \tanh (x) \left (a \cosh ^2(x)\right )^{5/2}+\frac {4}{5} a \left (\frac {1}{3} \tanh (x) \left (a \cosh ^2(x)\right )^{3/2}+\frac {2}{3} a \text {sech}(x) \sqrt {a \cosh ^2(x)} \int \sin \left (i x+\frac {\pi }{2}\right )dx\right )\right )\) |
\(\Big \downarrow \) 3117 |
\(\displaystyle \frac {1}{7} \tanh (x) \left (a \cosh ^2(x)\right )^{7/2}+\frac {6}{7} a \left (\frac {1}{5} \tanh (x) \left (a \cosh ^2(x)\right )^{5/2}+\frac {4}{5} a \left (\frac {1}{3} \tanh (x) \left (a \cosh ^2(x)\right )^{3/2}+\frac {2}{3} a \tanh (x) \sqrt {a \cosh ^2(x)}\right )\right )\) |
((a*Cosh[x]^2)^(7/2)*Tanh[x])/7 + (6*a*(((a*Cosh[x]^2)^(5/2)*Tanh[x])/5 + (4*a*((2*a*Sqrt[a*Cosh[x]^2]*Tanh[x])/3 + ((a*Cosh[x]^2)^(3/2)*Tanh[x])/3) )/5))/7
3.2.21.3.1 Defintions of rubi rules used
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[(-Cot[e + f*x ])*((b*Sin[e + f*x]^2)^p/(2*f*p)), x] + Simp[b*((2*p - 1)/(2*p)) Int[(b*S in[e + f*x]^2)^(p - 1), x], x] /; FreeQ[{b, e, f}, x] && !IntegerQ[p] && G tQ[p, 1]
Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Sin[e + f*x]^ n)^FracPart[p]/(Sin[e + f*x]/ff)^(n*FracPart[p])) Int[ActivateTrig[u]*(Si n[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] && !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) / ; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
Time = 0.43 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.53
method | result | size |
default | \(\frac {a^{4} \cosh \left (x \right ) \sinh \left (x \right ) \left (5 \cosh \left (x \right )^{6}+6 \cosh \left (x \right )^{4}+8 \cosh \left (x \right )^{2}+16\right )}{35 \sqrt {a \cosh \left (x \right )^{2}}}\) | \(38\) |
risch | \(\frac {a^{3} {\mathrm e}^{8 x} \sqrt {a \left (1+{\mathrm e}^{2 x}\right )^{2} {\mathrm e}^{-2 x}}}{896+896 \,{\mathrm e}^{2 x}}+\frac {7 a^{3} {\mathrm e}^{6 x} \sqrt {a \left (1+{\mathrm e}^{2 x}\right )^{2} {\mathrm e}^{-2 x}}}{640 \left (1+{\mathrm e}^{2 x}\right )}+\frac {7 a^{3} {\mathrm e}^{4 x} \sqrt {a \left (1+{\mathrm e}^{2 x}\right )^{2} {\mathrm e}^{-2 x}}}{128 \left (1+{\mathrm e}^{2 x}\right )}+\frac {35 a^{3} {\mathrm e}^{2 x} \sqrt {a \left (1+{\mathrm e}^{2 x}\right )^{2} {\mathrm e}^{-2 x}}}{128 \left (1+{\mathrm e}^{2 x}\right )}-\frac {35 \sqrt {a \left (1+{\mathrm e}^{2 x}\right )^{2} {\mathrm e}^{-2 x}}\, a^{3}}{128 \left (1+{\mathrm e}^{2 x}\right )}-\frac {7 a^{3} {\mathrm e}^{-2 x} \sqrt {a \left (1+{\mathrm e}^{2 x}\right )^{2} {\mathrm e}^{-2 x}}}{128 \left (1+{\mathrm e}^{2 x}\right )}-\frac {7 a^{3} {\mathrm e}^{-4 x} \sqrt {a \left (1+{\mathrm e}^{2 x}\right )^{2} {\mathrm e}^{-2 x}}}{640 \left (1+{\mathrm e}^{2 x}\right )}-\frac {a^{3} {\mathrm e}^{-6 x} \sqrt {a \left (1+{\mathrm e}^{2 x}\right )^{2} {\mathrm e}^{-2 x}}}{896 \left (1+{\mathrm e}^{2 x}\right )}\) | \(262\) |
Leaf count of result is larger than twice the leaf count of optimal. 817 vs. \(2 (56) = 112\).
Time = 0.28 (sec) , antiderivative size = 817, normalized size of antiderivative = 11.35 \[ \int \left (a \cosh ^2(x)\right )^{7/2} \, dx=\text {Too large to display} \]
1/4480*(70*a^3*cosh(x)*e^x*sinh(x)^13 + 5*a^3*e^x*sinh(x)^14 + 7*(65*a^3*c osh(x)^2 + 7*a^3)*e^x*sinh(x)^12 + 28*(65*a^3*cosh(x)^3 + 21*a^3*cosh(x))* e^x*sinh(x)^11 + 7*(715*a^3*cosh(x)^4 + 462*a^3*cosh(x)^2 + 35*a^3)*e^x*si nh(x)^10 + 70*(143*a^3*cosh(x)^5 + 154*a^3*cosh(x)^3 + 35*a^3*cosh(x))*e^x *sinh(x)^9 + 35*(429*a^3*cosh(x)^6 + 693*a^3*cosh(x)^4 + 315*a^3*cosh(x)^2 + 35*a^3)*e^x*sinh(x)^8 + 8*(2145*a^3*cosh(x)^7 + 4851*a^3*cosh(x)^5 + 36 75*a^3*cosh(x)^3 + 1225*a^3*cosh(x))*e^x*sinh(x)^7 + 7*(2145*a^3*cosh(x)^8 + 6468*a^3*cosh(x)^6 + 7350*a^3*cosh(x)^4 + 4900*a^3*cosh(x)^2 - 175*a^3) *e^x*sinh(x)^6 + 14*(715*a^3*cosh(x)^9 + 2772*a^3*cosh(x)^7 + 4410*a^3*cos h(x)^5 + 4900*a^3*cosh(x)^3 - 525*a^3*cosh(x))*e^x*sinh(x)^5 + 35*(143*a^3 *cosh(x)^10 + 693*a^3*cosh(x)^8 + 1470*a^3*cosh(x)^6 + 2450*a^3*cosh(x)^4 - 525*a^3*cosh(x)^2 - 7*a^3)*e^x*sinh(x)^4 + 140*(13*a^3*cosh(x)^11 + 77*a ^3*cosh(x)^9 + 210*a^3*cosh(x)^7 + 490*a^3*cosh(x)^5 - 175*a^3*cosh(x)^3 - 7*a^3*cosh(x))*e^x*sinh(x)^3 + 7*(65*a^3*cosh(x)^12 + 462*a^3*cosh(x)^10 + 1575*a^3*cosh(x)^8 + 4900*a^3*cosh(x)^6 - 2625*a^3*cosh(x)^4 - 210*a^3*c osh(x)^2 - 7*a^3)*e^x*sinh(x)^2 + 14*(5*a^3*cosh(x)^13 + 42*a^3*cosh(x)^11 + 175*a^3*cosh(x)^9 + 700*a^3*cosh(x)^7 - 525*a^3*cosh(x)^5 - 70*a^3*cosh (x)^3 - 7*a^3*cosh(x))*e^x*sinh(x) + (5*a^3*cosh(x)^14 + 49*a^3*cosh(x)^12 + 245*a^3*cosh(x)^10 + 1225*a^3*cosh(x)^8 - 1225*a^3*cosh(x)^6 - 245*a^3* cosh(x)^4 - 49*a^3*cosh(x)^2 - 5*a^3)*e^x)*sqrt(a*e^(4*x) + 2*a*e^(2*x)...
Timed out. \[ \int \left (a \cosh ^2(x)\right )^{7/2} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.99 \[ \int \left (a \cosh ^2(x)\right )^{7/2} \, dx=\frac {1}{896} \, a^{\frac {7}{2}} e^{\left (7 \, x\right )} + \frac {7}{640} \, a^{\frac {7}{2}} e^{\left (5 \, x\right )} + \frac {7}{128} \, a^{\frac {7}{2}} e^{\left (3 \, x\right )} - \frac {35}{128} \, a^{\frac {7}{2}} e^{\left (-x\right )} - \frac {7}{128} \, a^{\frac {7}{2}} e^{\left (-3 \, x\right )} - \frac {7}{640} \, a^{\frac {7}{2}} e^{\left (-5 \, x\right )} - \frac {1}{896} \, a^{\frac {7}{2}} e^{\left (-7 \, x\right )} + \frac {35}{128} \, a^{\frac {7}{2}} e^{x} \]
1/896*a^(7/2)*e^(7*x) + 7/640*a^(7/2)*e^(5*x) + 7/128*a^(7/2)*e^(3*x) - 35 /128*a^(7/2)*e^(-x) - 7/128*a^(7/2)*e^(-3*x) - 7/640*a^(7/2)*e^(-5*x) - 1/ 896*a^(7/2)*e^(-7*x) + 35/128*a^(7/2)*e^x
Time = 0.26 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.10 \[ \int \left (a \cosh ^2(x)\right )^{7/2} \, dx=\frac {1}{4480} \, {\left (5 \, a^{3} e^{\left (7 \, x\right )} + 49 \, a^{3} e^{\left (5 \, x\right )} + 245 \, a^{3} e^{\left (3 \, x\right )} + 1225 \, a^{3} e^{x} - {\left (1225 \, a^{3} e^{\left (6 \, x\right )} + 245 \, a^{3} e^{\left (4 \, x\right )} + 49 \, a^{3} e^{\left (2 \, x\right )} + 5 \, a^{3}\right )} e^{\left (-7 \, x\right )}\right )} \sqrt {a} \]
1/4480*(5*a^3*e^(7*x) + 49*a^3*e^(5*x) + 245*a^3*e^(3*x) + 1225*a^3*e^x - (1225*a^3*e^(6*x) + 245*a^3*e^(4*x) + 49*a^3*e^(2*x) + 5*a^3)*e^(-7*x))*sq rt(a)
Timed out. \[ \int \left (a \cosh ^2(x)\right )^{7/2} \, dx=\int {\left (a\,{\mathrm {cosh}\left (x\right )}^2\right )}^{7/2} \,d x \]